Question: Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{a^2 + 10a}{a^2 + 20a + 100}$
First factor the expressions in the numerator and denominator. $ \dfrac{a^2 + 10a}{a^2 + 20a + 100} = \dfrac{(a)(a + 10)}{(a + 10)(a + 10)} $ Notice that the term $(a + 10)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(a + 10)$ gives: $r = \dfrac{a}{a + 10}$ Since we divided by $(a + 10)$, $a \neq -10$. $r = \dfrac{a}{a + 10}; \space a \neq -10$